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March 2020 Minimal surfaces in the $3$-sphere by stacking Clifford tori
David Wiygul
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J. Differential Geom. 114(3): 467-549 (March 2020). DOI: 10.4310/jdg/1583377214

Abstract

Extending work of Kapouleas and Yang, for any integers $N \geq 2, {k , \ell} \geq 1$, and m sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus ${k \ell m}^2 (N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by ${k \ell m}^2(N-1)$ small catenoidal tunnels, with ${k \ell m}^2$ tunnels joining each pair of neighboring tori. In the large-$m$ limit for fixed $N$, $k$, and $\ell$, the corresponding surfaces converge to $\mathbb{T}$ counted with multiplicity $N$.

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David Wiygul. "Minimal surfaces in the $3$-sphere by stacking Clifford tori." J. Differential Geom. 114 (3) 467 - 549, March 2020. https://doi.org/10.4310/jdg/1583377214

Information

Received: 15 March 2015; Published: March 2020
First available in Project Euclid: 5 March 2020

zbMATH: 07179185
MathSciNet: MR4072204
Digital Object Identifier: 10.4310/jdg/1583377214

Rights: Copyright © 2020 Lehigh University

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Vol.114 • No. 3 • March 2020
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